Frodo's was far more perilous. The entire future of Middle-earth rested on the Ring being destroyed (not Smaug being killed).

As I was writing this reply, I was going to say that Frodo and Sam's journey was relatively more simple than Bilbo's. But after writing down the events, I've come to the conclusion that both journeys were as dangerous as the other. Both Bilbo and Frodo got into a lot of danger:

While both Bilbo and Frodo held the Ring, Frodo had it when Sauron was actively searching for it and the Ring had "heard its master's call". I figure that means that the Ring was more determined to exert its influence to get back to Sauron - an influence that grew stronger the closer Frodo got to Barad-dur, placing Frodo in greater danger of falling under Sauron's control.

Frodo's adventure was far, far more perilous. The weight of the world rested upon it, and he faced the nigh-impossible task of carrying the Ring as a burden rather than something which was fairly handy, as it was for Bilbo.

You just need to look at what is left of Frodo in comparison to Bilbo at the end of it: Frodo has been given a wound that will never fully heal, not to mention losing his finger. He realizes that "some hurts go too deep" and that "there is no going back". It effectively destroys his life in Middle-earth.

As others have said, Frodo is actively being pursued by enemies. Plus for a great bit of his journey, it's just him, Sam, and Gollum. On the other hand, Bilbo had a whole bad of dwarves for most of his journey.

murder of crows... congregation of alligators... cauldron of bats... kettle of hawks.... sleuth of bears (Bearlock????) a flutter of butterflies... a clowder of cats... a coalition of cheetahs.... a quiver of cobras... an intrusion of cockroaches... a consortium of crabs... a convocation of eagles... a business of ferrets, a skulk of foxes, a conspiracy of lemurs, a risk of lobsters, a maelstrom of salamanders, a shiver of sharks, an audience of squid, a pot of water buffalo, a tower of giraffes, a thunder of hippos, a horde of hamsters, and a cackle of hyenas...

for real... I couldn't make this stuff up... http://www.buzzfeed.com/donnad/36-bizarre-group-names-for-animals

I mean really, Bilbo had to fight trolls and dragons, while Frodo was fighting for his soul and those of all Middle-Earth! Bilbo never saw a ringwraith, or an Uruk-Hai, and he was certainly never stabbed with a Morgul Blade!

Frodo was nearly eaten twice, had to save all of Middle-Earth, nearly lost his mind and soul to the Ring, plus a ton of other things that for the sake of time and the fact that various others have already posted I will not mention!